It's hard to correct wrongs after the fact, but as I've learned from Ruso's times being removed from MKWRS, it's never too late to do so.
In my opinion: keep Fox's time,
maybe keep Hibiki's time, remove all others.
Reasoning: Fox's time is the earliest case out of these 4, he has the most credibility out of all of these players (with WRs going back as far as 2013), and he's given a good enough reason for his actions. The game modification isn't intentionally advantageous either. While Fox accepts the repercussions of using game mods in TTs, I think that if this were to be the only exception ever made, it'd be a good one.
Hibiki's WR was set nearly a year later and does include the My Stuff warning, which makes his case less compelling for me. Even if he hadn't known the PP's ruleset, he still had a warning right in front of him. On top of that, there is obviously no ghost for the run, which I personally see as a necessary piece of credibility for a WR in this day and age.
However, Hibiki does have legitimate 2:05 DKM ghosts, and has other times on MKLB with ghost proof as well. He's clearly a capable player just like Fox, but we can't be sure about his reasons for using custom music in his WR, especially with the warning on the bottom of the screen. I say, if the community trusts the run enough, keep it. But from a ruleset standpoint, this run should be retroactively removed.
Mel and Kasper are both online players and their runs were set long after both of these cases. I don't doubt that they were aware of the custom music ban, and TT'd with it anyway because they didn't care. Nothing is stopping them from redoing those runs, as they are both extremely capable and credible players. But even if no "official" ruleset was determined at that point in time, I think the message would have been clear enough for everyone.
For
Cole's post here, I support 1.0 and 1.1 - I strongly think 2.0 only complicates things. There is room for error with different updaters having different opinions on modded WRs. On top of that, the additional "strike" players would incur can just be easily avoided by enforcing the "no mods, ever" rule from the get go. I get the idea - it's nice to make exceptions for fluke runs by credible players because the run is a WR after all. But the precedent has long been established and as such I see no excuse for players to have this lifeline.
To address Firestone's point about modded ROMs in SM64: I think it's a good point but it always comes back to consensus. It's what the community agrees on that should be the norm, for better or worse. You could argue CTGP isn't 100% equivalent to the way MKW was played in 2008, but the community has accepted it as the new norm because it's a large improvement over vanilla MKW with features that enhance the experience without significantly changing the game. Rulesets should always be changed and updated to what the community agrees on, but in the particular case of custom textures and music, the consensus has been established for so long that I personally wouldn't accept any modded WRs from this point on.
To close out my post, the PP is no longer (or shouldn't be) the central authority for TTs. I hope that authority shifts to MKWRS & MKLB, because it's people like Arvo who make most of the good decisions nowadays. They care about the game & community, and understand all the fundamental issues from every point of view. Having open discussions like this just shows how fair and unbiased these people are - they want you to contribute to the final decision. Revising past issues is a controversial but needed task, and these new gen players always come through with some good points to settle the problem.